Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{a^2 - 2a - 80}{a^2 + 5a - 24}$
First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - 2a - 80}{a^2 + 5a - 24} = \dfrac{(a - 10)(a + 8)}{(a - 3)(a + 8)} $ Notice that the term $(a + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 8)$ gives: $q = \dfrac{a - 10}{a - 3}$ Since we divided by $(a + 8)$, $a \neq -8$. $q = \dfrac{a - 10}{a - 3}; \space a \neq -8$